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 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.: Year: Volume: Issue: Page: Find

 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2019, Volume 19, Issue 1, Pages 4–15 (Mi isu785)

Scientific Part
Mathematics

Approximation of continuous $2\pi$-periodic piecewise smooth functions by discrete Fourier sums

G. G. Akniev

Dagestan Scientific Center RAS, 45 M. Gadzhieva St., 367025 Makhachkala, Russia

Abstract: Let $N$ be a natural number greater than $1$. Select $N$ uniformly distributed points $t_k = 2\pi k / N + u$ $(0 \leq k \leq N - 1)$, and denote by $L_{n,N}(f)=L_{n,N}(f,x)$ $(1\leq n\leq N/2)$ the trigonometric polynomial of order $n$ possessing the least quadratic deviation from $f$ with respect to the system $\{t_k\}_{k=0}^{N-1}$. Select $m+1$ points $-\pi=a_{0}<a_{1}<\ldots<a_{m-1}<a_{m}=\pi$, where $m\geq 2$, and denote $\Omega = \{a_i\}_{i=0}^{m}$. Denote by $C_{\Omega}^{r}$ a class of $2\pi$-periodic continuous functions $f$, where $f$ is $r$-times differentiable on each segment $\Delta_{i}=[a_{i},a_{i+1}]$ and $f^{(r)}$ is absolutely continuous on $\Delta_{i}$. In the present article we consider the problem of approximation of functions $f\in C_{\Omega}^{2}$ by the polynomials $L_{n,N}(f,x)$. We show that instead of the estimate $|f(x)-L_{n,N}(f,x)| \leq c\ln n/n$, which follows from the well-known Lebesgue inequality, we found an exact order estimate $|f(x)-L_{n,N}(f,x)| \leq c/n$ ($x \in \mathbb{R}$) which is uniform with respect to $n$ ($1 \leq n \leq N/2$). Moreover, we found a local estimate $|f(x)-L_{n,N}(f,x)| \leq c(\varepsilon)/n^2$ ($|x - a_i| \geq \varepsilon$) which is also uniform with respect to $n$ ($1 \leq n \leq N/2$). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.

Key words: function approximation, trigonometric polynomials, Fourier series.

DOI: https://doi.org/10.18500/1816-9791-2019-19-1-4-15

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Bibliographic databases:

UDC: 517.521.2
Accepted:28.11.2018
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Citation: G. G. Akniev, “Approximation of continuous $2\pi$-periodic piecewise smooth functions by discrete Fourier sums”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 19:1 (2019), 4–15

Citation in format AMSBIB
\Bibitem{Akn19} \by G.~G.~Akniev \paper Approximation of continuous $2\pi$-periodic piecewise smooth functions by discrete Fourier sums \jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. \yr 2019 \vol 19 \issue 1 \pages 4--15 \mathnet{http://mi.mathnet.ru/isu785} \crossref{https://doi.org/10.18500/1816-9791-2019-19-1-4-15} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000461458700001}